Empirical Spectral Distribution of a Matrix Under Perturbation

  • Florent Benaych-Georges
  • Nathanaël EnriquezEmail author
  • Alkéos Michaïl


We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.


Random matrices Perturbation theory Wigner matrices Band matrices Hilbert transform Spectral density 

Mathematics Subject Classification (2010)

15A52 60B20 47A55 46L54 



We thank Jean-Philippe Bouchaud, Guy David and Vincent Vargas for some fruitful discussions. We are also glad to thank the GDR MEGA for partial support.


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Authors and Affiliations

  1. 1.Université Paris DescartesParis Cedex 06France
  2. 2.Laboratoire de mathématiques d’Orsay (UMR 8628)Université Paris-SudOrsayFrance

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